Cross talking down about his former Pokémon—and about how he feels about Pokémon in general that he deems weak—can tug a few heartstrings.
The first encounter with Charmander is also very depressing - unlike its main series counterpart, who when left in the rain hid its tail under a leaf to prevent it from going out (if its tail goes out, it'll die), this Charmander was just sitting there fully exposed.
Sorrel's childhood memory of his parents' Luxray dying by his side as it protects his injured self from the cold. The way the scene is portrayed does not help at all; how child Sorrel was instantly relieved to see the Pokémon be there to help keep him warm, only to wake up learning that it had died, how the Luxray was smiling in death, and how Sorrel was sobbing for his deceased friend as he was being taken away; this is easily one of the heaviest scenes in the entire anime. This was the first time a dead Pokémon was shown in the anime that wasn't revived. No turning to stone or vanishing. The heaviness of it doesn't let up at all in the dubbed version, either.
Minor ones each time when Ash's Pokémon evolve, specifically how ecstatic and proud their trainer is of them.
Ash's little breakdown after he loses the battle with Cross. He refuses to accept that he lost, and to a heartless trainer at that. He's so upset that he mutters how he could have won using another Pokémon, or that he wish he got another starter Pokémon that wasn't Pikachu. Saying all of this upsets him even further, causing him to run off into the forest alone, to find himself defeated and helpless without his friends.
Ash releasing Butterfree, like in the show, is also a big one. Might be more or less impactful since the show's theme song doesn't play in the background here. Also depends on how you feel about the line deliveries.
A minor one, after Cross' HeelFace Turn, when he tackles his Lycanroc—then corrupted by Marshadow—before it attacks Ash and friends with Stealth Rock. The brainwashed Wolf Pokémon bites his trainer in blind retaliation; Cross takes the pain while yelling at the Pokémon to remember when it bit him on the first day they met, in order to help Lycanroc to snap out of it. The affectionate lick it gives its trainer after it comes to is on the heartwarming side of a tearjerker.
Ash's subsequent death from the Pokémon's attacks. A little reminiscent of another time Ash died, with it being due to Pokémon, and Pikachu's slowly-building anguish when he comes to realization of the situation. Pikachu uncontrollably sobbing for his friend isn't any easier to watch than the other time it happened. Might be even worse, since this time Ash just flat-out dissolves, so there's no body for Pikachu or any other Pokémon to cry on and bring him back that way.
Depending on if you see it as narmy, Pikachu speaking to Ash before his death is the icing on the cake.
Not to mention the music playing in the background: a sad Piano rendition of the original theme. It just gets you right in the feels.
Pikachu breaking down into tears and then outright having an electric Angst Nukeexplosion of grief so powerful it clears the sky and blasts every Pokémon within its radius away like ragdolls. Yikes, and we thought Pikachu becoming a living defibrillator in the first movie was hard to watch.
The opening montage of Ash and Pikachu journeying and training through Kanto, with the original theme song remix playing in the background can bring tears of nostalgia.
Similarly, the narrator's monologue at the end of the movie, raving about the countless, wonderful adventures Ash and Pikachu were bound to have, just like in the first episode, though this time showing a few locales from previous movies including Shamouti Island, Crown City, Alamos Town, Eindoak Town, and Allearth Forest followed by Alola. The famous "Avant" theme that's commonly heard at the "World of Pokémon" montages in the regular movies makes it all the better.
During the credits, all of Ash's friends from the main anime pre-Alola appear, also stirring some nostalgia to those who watched the anime that long and are very happy to see them once more.
Serena in particular could cause some emotions considering what happened at the end of the XY series, not to mention her English voice actress Haven Paschall singing the ending theme.